Three digit numbers with repeated digits

CAT 2020 Slot 3 · QA · Easy · Number theory

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How many integers in the set {100,101,102,…,999}\{100, 101, 102, \dots, 999\} have at least one digit repeated?

Answer

252

Explanation

Total 3-digit numbers from 100 to 999 = 999−100+1=900999 - 100 + 1 = 900.

Number of 3-digit integers with ALL digits distinct:

  • First digit (hundreds): 9 choices (1−91-9)
  • Second digit (tens): 9 choices (0−90-9 excluding hundreds digit)
  • Third digit (units): 8 choices (0−90-9 excluding first two digits) Total distinct digit numbers = 9×9×8=6489 \times 9 \times 8 = 648.

Numbers with at least one digit repeated = 900−648=252900 - 648 = 252.

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